The Carry Trade: Risks and Drawdowns

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6 foreign currency money market, or by borrowing the appropriate amount of foreign currency to invest one dollar in the dollar money market. When interest rate parity holds, if i t > i \$ t, F t < S t, the foreign currency is at a discount in the forward market. Conversely, if i t < i \$ t, F t > S t, and the foreign currency is at a premium in the forward market. Thus, the carrytrade can also be implemented by going long (short) in the foreign currency in the forward market when the foreign currency is at a discount (premium)in terms of the dollar. Let w t be the amount of foreign currency bought in the forward market. The dollar payoff to this strategy is z t+1 = w t (S t+1 F t ) (2) To scale the forward positions to be either long or short in the forward market an amount of dollars equal to one dollar in the spot market as in equation (1), let w t = { 1 F t (1 + i \$ t ) 1 F t (1 + i \$ t ) } if F t < S t (3) if F t > S t When covered interest rate parity holds, and in the absence of transaction costs, the forward market strategy for implementing the carry trade in equation (2) is exactly equivalent to the carry trade strategy in equation (1). Unbiasedness of forward rates and uncovered interest rate parity imply that carry trade profits should average to zero. Uncovered interest rate parity ignores the possibility that changes in the values of currencies are exposed to risk factors, in which case risk premiums can arise. To incorporate risk aversion, we need to examine pricing kernels. 2.2 Pricing Kernels One of the fundamentals of no-arbitrage pricing is that there is a dollar pricing kernel or stochastic discount factor, M t+1, that prices all dollar returns, R t+1, from the investment of one dollar: E t [M t+1 R t+1 ] = 1 (4) Because implementing the carry trade in the forward market does not require an investment at time t, the no-arbitrage condition is E t (M t+1 z t+1 ) = 0 (5) 6

11 returns realized at time t + 1. Then, the IGARCH model is h ij t = α(r i tr j t ) + (1 α)h ij t 1 (9) where we treat the product of the returns as equivalent to the product of the innovations in the returns. We set α = 0.06, as suggested in RiskMetrics (1996). To obtain the monthly covariance matrixes we multiply the daily IGARCH estimates of H t by 22. We use these conditional covariance matrixes in two strategies. In the first strategy, we target a fixed monthly standard deviation of 5%/ 12, which corresponds to an approximate annualized standard deviation of 5%, and we adjust the dollar scale of the EQ and SPD carry trades accordingly. These strategies are labeled EQ-RR and SPD-RR to indicate risk rebalancing. In the second strategy, we use the conditional covariance matrixes in successive one-month mean-variance maximizations. Beginning with the analysis of Meese and Rogoff (1983), it is often argued that expected rates of currency appreciation are essentially unforecastable. Hence, we take the vector of interest differentials, here labeled µ t, to be the conditional means of the carry trade returns, and we take positions wt OP T = ω t Ht 1 µ t, where H t is the conditional covariance matrix, and ω t is a scaling factor. We label this strategy OPT. As with the risk-rebalanced strategies, we target a constant annualized standard deviation of 5%. Thus, the weights in our portfolio are w OP T t = ( 0.05/ 12 ) ( ) µ t Ht Ht 1 µ t (10) µ t If the models of the conditional moments are correct, the conditional Sharpe ratio would equal ( µ th 1 t µ t ) Table 1 reports the first four moments of the various carry trade strategies as well as their Sharpe ratios and first order autocorrelations. Standard errors are in parenthesis and are based on Hansen s (1982) Generalized Method of Moments, as explained in Appendix B. 9 For the full sample, the carry trades for the USD-based investor have statistically significant average annual returns ranging from 2.10% (0.47) for the OPT strategy, to 3.96% (0.91) for the EQ strategy, and to 6.60% (1.31) for the SPD strategy. The strategies also have impressive Sharpe ratios, which range from 0.78 (0.19) for the EQ strategy to 1.02 (0.19) for the SPD- RR strategy. As Brunnermeier, Nagel, and Pedersen (2009) note, each of these strategies is 8 Ackermann, Pohl, and Schmedders (2012) also use conditional mean variance modeling so their positions are also proportional to Ht 1 µ t, but they target a constant mean return of 5% per annum. Hence, their positions satisfy w AP S µ t. While their conditional Sharpe ratio is also ( µ th 1 ) 0.5, µ t their t = (0.05/12) H 1 µ t H 1 t µ t t scaling factor responds more aggressively to perceived changes in the conditional Sharpe ratio than ours. 9 Throughout the paper, when we discuss estimated parameters, standard errors will be in parentheses and t-statistics will be in square brackets. 11 t

13 5 Carry-Trade Exposures to Risk Factors This section examines whether the average returns to the carry trades described above can be explained by exposures to a variety of risk factors. We include equity market, foreign exchange market, bond market, and volatility risk factors. In each case we run a regression of a carry-trade return, R t, on returns on assets or portfolios, F t, that represent the sources of risks, as in R t = α + β F t + ε t (11) Because we use returns as risk factors, the constant term in the regression, α, measures the average performance of the carry trade that is not explained by unconditional exposure of the carry trades to the market traded risks included in the regression. 5.1 Equity Market Risks Table 3 presents the results of regressions of our carry-trade returns on the three Fama-French (1993) equity market risk factors: the excess market return, R m,t, proxied by the return on the value-weighted NYSE, AMEX, and NASDAQ markets over the T-bill return; the return on a portfolio of small market capitalization stocks minus the return on a portfolio of big stocks, R SMB,t ; and the return on a portfolio of high book-to-market stocks minus the return on a portfolio of low book-to-market stocks, R HML,t. 11 Although the Fama-French (1993) factors exhibit some modest explanatory power with nine of the 15 coefficients having t-statistics that are greater than 1.88, the overall impression is that these equity market factors essentially leave most of the average returns of the carry trades unexplained, as the exposures to the risk factors are quite small which allows that constants in the regressions to range from 1.83% with a t-statistic of 3.72 for the OPT strategy, to 5.55% with a t-statistic of 4.84 for the SPD-RR strategy. The largest R 2 is also only.05. These equity market risk factors clearly do not explain the average carry trade returns. 5.2 Pure FX Risks Table 4 presents the results of regressions of our carry-trade portfolio returns on the two pure foreign exchange market risk factors proposed by Lustig, Roussanov, and Verdelhan (2011) who sort 35 currencies into six portfolios based on their interest rates relative to the dollar interest rate, with portfolio one containing the lowest interest rate currencies and portfolio six containing the highest interest rate currencies. Their two risk factors are the average 11 The Fama-French risk factors were obtained from Kenneth French s web site which also describes the construction of these portfolios. 13

17 Subtracting the currency positions in the EQ0 strategy from those in EQ strategy gives the positions in EQ-minus, which goes long (short) the dollar when the dollar interest rate is higher (lower) than the median interest rate but only against currencies with interest rates between the median and dollar interest rates. The exact positions of the portfolio are the following: If i \$ t < med { i k t }, then y j t = 2 N t 0 if i j t > med { } i k t if i j t = med { } i k t if i \$ t < i j t < med { } i k t 0 if i j t i \$ t 1 N t If i \$ t > median { } i k t, then y j t = 2 N t 0 if i j t > i \$ t if i j t = med { } i k t if med { } i k t < i j t i \$ t 0 if i j t med { } i k t 1 N t The EQ0 and EQ-minus portfolios decompose the EQ carry trade into two components: a dollar neutral component and a dollar component. Column 3 of Panels A and B in Table 8 presents the first four moments of the EQ-minus strategy. which has statistically significant average annual returns in both samples, 2.35 (0.66) for the full sample and 2.11 (0.92) for the later sample. The Sharpe ratio of the EQ-minus strategy is higher than that of the EQ0 strategy in the full sample, 0.61 versus 0.49; but in later half of the sample, the EQ-minus strategy has a slightly lower Sharpe ratio, 0.49 versus 0.52, than the EQ0 strategy. Given the standard errors, one could argue that the Sharpe ratios are the same. Skewness of EQ-minus is quite negative (0.44) and (0.45) for the full sample and the later sample, respectively, although we note that the estimates of skewness are actually statistically insignificant due to the large standard error in both samples. In terms of the Sharpe ratio and skewness, the EQ-minus strategy appears no better than the EQ0 strategy. Nevertheless, the EQ-minus strategy has a correlation of -.11 with the EQ0 strategy, and the following results illustrate that the EQ-minus strategy also differs significantly from the EQ0 strategy in its risk exposures. Column 3 of Table 9 presents the results of regressions of the EQ-minus returns on the three Fama-French (1993) equity market risk factors. Unlike EQ0, only the SMB factor shows any explanatory power for the EQ-minus returns. The constant in the regression is 2.36% 17

18 with a t-statistic of The R 2 is.04. The equity market risk factors clearly do not explain the average returns to the EQ-minus strategy. Columns 1 to 3 of Table 10 present regressions of the returns to the EQ strategy and its two components, EQ0 and EQ-minus, on the equity market excess return and two bond market risk factors. Similar to our previous findings, the market excess return and the bond risk factors have significant explanatory power for the returns of EQ0 and a relatively high R 2 of.12. By comparison, none of the risk factors has any significant explanatory power for the returns on the EQ-minus strategy, and the resulting R 2 is only.02. Finally, Columns 1 to 3 of Table 11 present regressions of the returns of EQ and its two components, EQ0 and EQ-minus, on the two FX risk factors. The two factor FX model completely explains the average returns of the EQ0 strategy while explaining only 25% of the average returns of EQ-minus. The constant term in the EQ-minus regression also is significant with a value of 1.49% and a t-statistic of 1.86 in the FX two factor model. In summary, these results suggest that for the G10 currencies, conditional dollar exposure contributes more to the carry trade puzzle than does the non-dollar component. 6.2 The Pure Dollar Factor As noted above, the EQ-minus strategy goes long (short) the dollar when the dollar interest rate is above (below) the G10 median interest rate. It has the nice property of complementing the EQ0 strategy to become the commonly studied equally weighted carry trade. However, the other leg of the EQ-minus portfolio goes short (long) the currencies with interest rates between the G10 median rate and the dollar interest rate. It takes positions in relatively few currencies and thus seems under-diversified, which could explain its large kurtosis. Since the results just presented indicate that the abnormal returns of EQ hinge on the conditional dollar exposure, which is distinct from carry, we now expand the other leg of EQ-minus to all foreign currencies. We thus create the following strategy EQ-USD: { + y j 1 N t = if med { } } i k t > i \$ t 1 if med { } i k N t i \$ t The EQ-USD strategy focuses on the conditional exposure of the dollar. It goes long (short) the dollar against all nine foreign currencies when the dollar interest rate is higher (lower) than the global median interest rate. The fourth columns of Panels A and B of Table 8 present the first four moments of the returns to the EQ-USD strategy for the full sample and the sample for which equity volatility is available. We find that EQ-USD has statistically significant average annual returns in both samples, 5.54% (1.37) for the full sample and 5.21% 18

20 Lettau, Maggiori, and Weber (2014) modify the downside market risk model of Ang, Chen, and Xing (2006) and argue that the exposure of the carry trade to the return on the market is larger, conditional on the market return being down. In their empirical analysis, Lettau, Maggiori, and Weber (2014) define the downside market return, which we denote Rm,t, as equal to the market return when the market return is one standard deviation below the average market return and zero otherwise. 13 Lettau, Maggiori, and Weber (2014) find that the down-market-beta differential between the high and low interest rate sorted portfolios combined with a high price of down market risk is sufficient to explain the average returns to the carry trade. To examine whether the Rm,t risk factor explains our G10 carry trade strategies, we investigate whether our six currency portfolios have significant betas with respect to Rm,t and whether these downside betas are significantly different from the standard betas We show that our portfolios as well as other currency portfolios more generally, have statistically insignificant beta differentials which invalidates the explanation of Lettau, Maggiori, Weber (2014) for these carry trades. In their econometric analysis, Lettau, Maggiori, and Weber (2014) run separate univariate OLS regressions of portfolio returns, R t, on R m,t and Rm,t to define the risk exposures, β and β. Then, they impose that the price of R m,t risk is the average return on the market, E (R m,t ), and they use a cross-sectional regression to estimate a separate price of risk for Rm,t, denoted λ, which is necessary because Rm,t is not a traded risk factor. The unconditional expected return on an asset is therefore predicted to be E (R t ) = βe (R m,t ) + (β β)λ To test this explanation of our carry trades we first run a bivariate regression of a carry trade return on a constant; an indicator dummy variable, I, that is one when R m,t is non-zero and zero otherwise; and the two risk factors, R m,t and R m,t. Thus, we estimate R t = α 1 + α 2 I + β 1 R m,t + β 2 R m,t + ε t (12) In equation (12), β 1 measures the asset s basic exposure to the market excess return given that there is additional adjustment for when the market is in the downstate; and β 2 measures the asset s additional exposure to the market excess return when the market is in the downstate, that is β 2 = (β β 1 ). We then test the significance of β 2 using heteroskedasticity and 13 Because the definition of Rm,t used by Lettau, Maggiori, and Weber (2014) seemed arbitrary to us, we also considered two alternative more intuitive definitions which are the market return being less than the unconditional average market return and the market return being negative. The results are similar to the reported results and are available in an internet appendix. 20

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